PRICE OFFAME: While Angelina Jolie and Brad Pitt project a glamorous image jetting off to red-carpet premieres, their six kids, including (from left) Maddox, Zahara and Shiloh, spend so much time with caretakers that one London hotel employee heard Shiloh call a nanny “Mommy.”Ramey Photo

Hollywood’s most glamorous couple is headed for Splitsville, according to a blockbuster book, “Brangelina: The Untold Story of Brad Pitt and Angelina Jolie” (Transit), out Tuesday. Author Ian Halperin, who spoke to 900 sources to build a “true picture” of the pair, talked exclusively to The Post about his most sensational findings.

It was 2004, just weeks before shooting was to begin on the action flick “Mr. and Mrs. Smith,” when Angelina Jolie happened to visit the set of the hit show “Friends.”

“Brad is so excited to be working with you,” Jennifer Aniston told her husband’s new co-star. “I hope you guys have a really good time.”

They certainly did. Less than a year later, Brad Pitt and Jolie would pose for W magazine in a spread called “Domestic Bliss.” Real-life romance soon followed, as modern Hollywood’s most famous love triangle played out in the tabloids. Eventually, Pitt would leave Aniston and take up with Jolie, jetting to red carpets and orphanages around the world.

It was the electrifying love-making that brought Brangelina together, says Halperin, who followed the couple for five years to prepare to write his book. “Brad had never had such incredible sex,” Halperin says. “My sources say that they sometimes spent 18 to 20 hours a day in bed. But sex eventually wears off.”

ESCALATING TENSIONS

Now, Halperin claims, the couple “argues like cats and dogs — everything from politics to clothes the kids wear and Brad’s ‘atrocious habit,’ which includes pot smoking and alcohol.”

“I talked to several people in Brad’s family who wished they’d break up today. They’re scared for Brad’s safety and his well-being,” he says.

A limo driver who drove Brangelina frequently witnessed them “going at it in the back seat” twice and saw a “U-turn” in their relationship in 2007.

“I was more worried about his safety than hers,” the driver says. “She really flew off the handle, threatening him and lashing out at him. I can’t imagine how anyone would want to be with her every day, no matter how sexy she is. She has a temper like a cobra.”

A woman who worked with Pitt while he was filming “Troy” told Halperin: “They were definitely in love, but that’s where it gets tricky. If you want to know if they’re still together, the answer depends on when you ask the question.

“I’ve heard that they’ve broken up so many times it would make your head spin.

“There are apparently screaming matches, usually with her doing the screaming. Nobody really witnesses that part of it — they just see him leaving in a huff. But then he ends up coming back again, and nobody knows what happened to bring him back.”

CHEATING AND DRUGS

Pitt may have cheated on Jolie, according to Halperin’s book.

The most persistent story that crops up time and again involves Pitt and a beautiful Sudanese model, Amma, whom he met at a benefit for Darfur at the 2007 Cannes Film Festival. He was seen with her at least one more time. Jolie has allegedly accused Pitt of cheating on her on more than one occasion, which is one of the reported sources of tension between the two, Halperin writes.

Another tension is Pitt’s pot use, the book states. According to a well-known Hollywood journalist, “[Pitt and Aniston] spent much of their married life stoned. But whereas he was chilled out, she could get pretty paranoid.”

A woman who works for an industry publication says his drug use will give Jolie an easy way to explain their split, when it inevitably happens: “They’ll be split within 18 months, probably sooner,” she told Halperin.

“And here’s how it’s going to play out. They’re going to split amicably, work out some kind of arrangement with the kids, and everything will be very civilized. Then you’re going to see unidentified friends leak stories how Angelina couldn’t put up with Brad’s partying and drinking and his pot smoking. She’s worried about the kids, and she’s afraid it’s not the right environment for them. Maybe he’ll even take part of the blame and go along with that.

‘MANIPULATOR’ JOLIE

Halperin claims that Pitt’s biggest problem with Jolie is her controlling nature. “She won’t let [Pitt] out of the house alone,” he said. “She’s very guarded about him. I don’t think she’s evil, but she’s manipulative, deceiving and cunning.”

She has even spread lies about love rival Aniston to make herself look better in the public eye, Halperin claims.

It happened when Jolie first met and fell for Pitt on the set of “Mr. and Mrs. Smith.” At the time, he was still married to Aniston — and Jolie feared she’d be branded a homewrecker, Halperin said. So Jolie leaked a story about Aniston, claiming that the “Friends” star didn’t want children but wanted to focus on her career — and this was the reason her marriage had ended, the book states.

By now, Jolie was an accomplished veteran at what one publicist called “misdirection” — deflecting a crisis by changing the subject.

“It was Angelina who came up with the whole story about Jennifer not wanting to have kids,” a source connected to 20th Century Fox, which produced “Mr. and Mrs. Smith,” said. “She was convinced that is the story that would resonate with all these women who saw her as a man-stealing bitch.”

Pitt was furious when he heard about the mean-spirited whisper campaign that his new girlfriend was spreading about his ex-wife. He ordered Jolie to stop the rumors.

BABY HUNGER

Aniston, in fact, was thought to be pregnant twice during her marriage — both times ending in miscarriage, in 2003 and 2004, according to Halperin.

Pitt and Aniston even built a dream nursery at their Beverly Hills mansion.

She had already designated a large, light-filled space in the corner of the couple’s Beverly Hills mansion as a nursery for their expected baby, Halperin writes. Aniston supervised its transformation from beginning to end during the extensive renovation process.

“It wasn’t just a room where you’d stick a crib and a mobile,” a contractor said. “It was quite elaborate, and she had very specific ideas about it.”

“They never decorated it as I saw, though,” explained the contractor. “We’re talking 2003. Maybe they were waiting to know whether the baby would be a boy or a girl.”

THE NANNY DIARIES

Brangelina, of course, is now famous for their growing brood; they have six children, three adopted and three biological.

Although it is widely agreed the couple loves their children, they aren’t entirely honest about the number of people they’ve hired to help raise them.

Jolie and Pitt employ a “multicultural” team of nannies to care for their young brood, Halperin writes. In fact, the children often spend more time with those nannies than they do with their parents, who spend weeks at a time shooting films and on goodwill trips, sometimes with a child or two in tow but never with all of them.

A hotel employee at the Dorchester Hotel in London once overheard 3-year-old Shiloh refer to one of her nannies as “Mommy,” Halperin writes.

The employee also observed celebrity efforts to broadcast an image of a happy family.

“There’s sometimes as much of a production involved in avoiding the wrong shot as there is in making a movie,” she confided.

“Stars will never allow the nanny to be photographed with their kids. . . Sometimes you’ll even have the nanny crouched down in the back seat of a car. It’s quite comical.”